Ivancevic_Applied-Diff-Geom

240 Applied Differential Geometry: A Modern IntroductionOn the Lie algebra level, there is a corresponding linear map from theLie algebra of G to End(V ) preserving the Lie bracket [·, ·].If the homomorphism is in fact an monomorphism, the representationis said to be faithful.A unitary representation is defined in the same way, except that Gmaps to unitary matrices; the Lie algebra will then map to skew–Hermitianmatrices.Now, if G is a semisimple group, its finite–dimensional representationscan be decomposed as direct sums of irreducible representations. The irreduciblesare indexed by highest weight; the allowable (dominant) highestweights satisfy a suitable positivity condition. In particular, there exists aset of fundamental weights, indexed by the vertices of the Dynkin diagram ofG (see below), such that dominant weights are simply non–negative integerlinear combinations of the fundamental weights.If G is a commutative compact Lie group, then its irreducible representationsare simply the continuous characters of G. A quotient representationis a quotient module of the group ring.3.8.6.3 Root Systems and Dynkin DiagramsA root system is a special configuration in Euclidean space that has turnedout to be fundamental in Lie theory as well as in its applications. Also, theclassification scheme for root systems, by Dynkin diagrams, occurs in partsof mathematics with no overt connection to Lie groups (such as singularitytheory, see e.g., [Helgason (2001); Weisstein (2004); Wikipedia (2005)]).DefinitionsFormally, a root system is a finite set Φ of non–zero vectors (roots)spanning a finite–dimensional Euclidean space V and satisfying the followingproperties:(1) The only scalar multiples of a root α in V which belong to Φ are αitself and -α.(2) For every root α in V , the set Φ is symmetric under reflection throughthe hyperplane of vectors perpendicular to α.(3) If α and β are vectors in Φ, the projection of 2β onto the line throughα is an integer multiple of α.The rank of a root system Φ is the dimension of V . Two root systemsmay be combined by regarding the Euclidean spaces they span as mutually